Keywords

1. Introduction

where λ is a real parameter and q is real-valued function which has a singularity in (a, b). According to [1], an eigenvalue problem may be associate with (1.1) by imposing the boundary conditions

y(a)cosα-y′(a)sinα=0,α∈[0,π),

(1.2)

y(b)cosβ-y′(b)sinβ=0,β∈[0,π).

(1.3)

In [2], Atkinson obtained an asymptotic approximation of eigenvalues where y satisfies Dirichlet and Neumann boundary conditions in (1.1). Here, we find asymptotic approximation of eigenvalues for all boundary condition of the forms (1.2) and (1.3). To achieve this, we transform (1.1) to a differential equation all of whose coefficients belong to L1[a, b]. Then we employ a Prüfer transformation to obtain an approximation of the eigenvalues. In this way, many basic properties of singular problems can be inferred from the corresponding regular ones. In [3], Harris derived an asymptotic approximation to the eigenvalues of the differential Equation (1.1), defined on the interval [a, b], with boundary conditions of general form. But, he demands the condition, q∈Ll[a, b]. Atkinson and Harris found asymptotic formulae for the eigenvalues of spectral problems associated with linear differential equations of the form (1.1), where q(x) has a singularity of the form αx-kwith 1≤k<43 and 1≤k<32 in [2, 4] respectively. Harris and Race [5] generalized those results for the case 1 ≤ k < 2. In [6], Harris and Marzano derived asymptotic estimates for the eigenvalues of (1.1) on [0, a] with periodic and semi-periodic boundary conditions. The reader can find the related results in [7–10]. We consider q(x) = Cx-Kwhere 1 ≤ K < 2 and an asymptotic approximation to the eigenvalues of (1.1) with boundary conditions of general form. Our technique in this article follows closely the technique used in [2–5]. Let U = [a, 0) ∪ (0, b] and q∈L1,Loc(U). As Harris did in [[5], p. 90], suppose that there exists some real function f on [a, 0) ∪ (0, b] in ACLoc([a, 0) ∪ (0, b]) which regularizes (1.1) in the following sense. For f which can be chosen in Section 2, define quasi-derivatives, y[i] as follows:

y[0]:=y,y[1]:=y′+fy,

y is a solution of (1.1) with boundary conditions (1.2) and (1.3) if and only if

y[0]y[1]=-f1f′+q-f2-λfy[0]y[1]

(1.4)

The object of the regularization process is to chose f in such way that

f∈L1(a,b)and-F:=q-f2+f′∈L1(a,b).

(1.5)

Having rewritten (1.1) as the system (1.4), we observe that, for any solution y of (1.1) with λ > 0, according to [2, 4], we can define a function θ∈AC(a, b) by

tanθ=λ12yy[1].

When y[1] = 0, θ is defined by continuity [[5], p. 91]. It makes sense to mention that one can find full discussions and nice examples about the choice of f in [2, 4, 5]. Atkinson in [2] noticed that the function θ satisfies the differential equation

θ′=λ12-fsin(2θ)+λ-12Fsin2(θ).

(1.6)

Let λ > 0 and the n-th eigenvalue λnof (1.1-1.3), then according to [[1], Theorem 2], Dirichlet and non-Dirichlet boundary conditions can be described as bellow:

It follows from (1.5-1.6) that large positive eigenvalues of either the Dirichlet or non-Dirichlet problems over [a, b] satisfy

λ12=θ(b)-θ(a)(b-a)+O(1).

(1.7)

Our aim here is to obtain a formula like (1.7) in which the O(1) term is replaced by an integral term plus and error term of smaller order. We obtain an error term of oλ-N2(N≥1). To achieve this we first use the differential Equation (1.6) to obtain estimates for θ(b) - θ(a) for general λ as λ → ∞.

2. Statement of result

We define a sequence ξj(t) for j = 1, ..., N + 1, t∈ [a, b] by

ξ1(t):=∫0tf(s)+F(s)dsξj(t):=∫0t(f(s)+F(s))ξj-1(s)ds

(2.1)

and note that in view of f, F∈L(a, b),

ξj(t)≤cξj-1(t)fort∈[a,b],2≤j≤N+1

(2.2)

Suppose that for some N ≥ 1,

f′ξN+1,f2ξN,fFξN∈L[a,b];f(t)ξN+1(t)→0ast→0.

(2.3)

We define a sequence of approximating functions a

θ0(x):=θ(a)+λ12(x-a);

(2.4)

θj(0):=θ(0);

(2.5)

θj+1(x):=θ(a)+λ12(x-a)-∫axfsin(2θj(t))dt+λ-12∫axFsin2(θj(t))dt.

(2.6)

for j = 0, 1, 2, ... and for a ≤ x ≤ b. We measure the closeness of the approximation in the next result. Thus

Copyright

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